All practice questions

Question 61Question →

A laboratory vessel contains a liquid solution consisting of water and ethanol, where water makes up $\frac{3}{8}$ of the total volume. First, $\frac{1}{5}$ of the total volume of the solution is drained and replaced with an equal volume of pure ethanol. Next, $\frac{1}{4}$ of the resulting mixture is evaporated, removing water and ethanol in proportion to their presence. Finally, pure water is added to fill the vessel back to its original total volume. What fraction of the final solution is water? Express your answer as a decimal.

Show answer & explanation

Answer: 0.475

Answer

The fraction of the final solution that is water is 0.475 (or 19/40).

By following the multi-step fractional changes to the liquid volume, the remaining water prior to refilling is

\frac{9}{40}

of the original capacity. Refilling the missing

\frac{1}{4}

(or

\frac{10}{40}

) volume with pure water yields

\frac{19}{40} = 0.475

of the total solution as water.

Step-by-Step Solution

1

Track water content after the initial replacement.

Water fraction becomes

\frac{3}{10}

of the original volume.

Let the original total volume be

V

. Initially, water volume is

\frac{3}{8}V

. Removing

\frac{1}{5}

of the solution leaves

\frac{4}{5}

of the original solution, so the water volume becomes

\frac{4}{5} \times \frac{3}{8}V = \frac{3}{10}V

. Replacing the removed volume with pure ethanol brings total volume back to

V

, with water occupying

\frac{3}{10}V

.

2

Track water content after evaporation.

Water volume becomes

\frac{9}{40}V

and total solution volume becomes

\frac{3}{4}V

.

Evaporating

\frac{1}{4}

of the solution leaves

\frac{3}{4}

of the mixture intact. The remaining water volume is

\frac{3}{4} \times \frac{3}{10}V = \frac{9}{40}V

.

3

Calculate the final water fraction after refilling with pure water.

Final water volume is

\frac{19}{40}V = 0.475V

.

To restore the total volume from

\frac{3}{4}V

back to

V

, an amount equal to

V - \frac{3}{4}V = \frac{1}{4}V

of pure water is added. Adding this to the existing water gives

\frac{9}{40}V + \frac{1}{4}V = \frac{9}{40}V + \frac{10}{40}V = \frac{19}{40}V

. Dividing by total volume

V

yields

\frac{19}{40} = 0.475

.

Key Concept

Sequential fractional reduction and component tracking

Estimated Time:2m 30s

Question 62Question →

Two automated timers are set to chime at regular intervals. The first timer chimes every $12$ minutes, and the second timer chimes every $18$ minutes. If both timers chime simultaneously at 12:00 PM, how many minutes will pass before they next chime at the exact same time?

Show answer & explanation

Answer: 36

Answer

36 minutes

The time interval between simultaneous chimes is given by the least common multiple (LCM) of the two individual intervals,

12

minutes and

18

minutes. Using prime factorization,

12 = 2^2 \times 3

and

18 = 2 \times 3^2

. The LCM takes the highest power of each prime present:

2^2 \times 3^2 = 4 \times 9 = 36

. Thus,

36

minutes will elapse before the timers chime together again.

Step-by-Step Solution

1

Find the prime factorizations of 12 and 18

12 = 2^2 \times 3

and

18 = 2 \times 3^2

Expressing numbers as products of prime factors allows us to easily compute the least common multiple.

2

Calculate the Least Common Multiple (LCM)

\text{LCM}(12, 18) = 2^{\max(2,1)} \times 3^{\max(1,2)} = 2^2 \times 3^2 = 36

The next simultaneous occurrence of periodic events is given by the LCM of their periods.

Key Concept

Least Common Multiple (LCM) of periodic events

Question 63Question →

A community library received a donation of $120$ new books. On Monday, $\frac{3}{8}$ of the donated books were cataloged. On Tuesday, $\frac{1}{3}$ of the remaining uncataloged books were cataloged. How many of the donated books remain to be cataloged?

Show answer & explanation

Answer: 50

Answer

50 books remain to be cataloged.

First, find the number of books cataloged on Monday: 3/8 of 120 is 45 books. Subtracting 45 from 120 leaves 75 books uncataloged. On Tuesday, 1/3 of those remaining 75 books were cataloged, which is 25 books. Subtracting 25 from 75 gives 50 books remaining to be cataloged.

Step-by-Step Solution

1

Calculate the number of books cataloged on Monday.

45 books

Multiply the initial total of 120 books by the fraction cataloged on Monday, which is 3/8.

2

Calculate the uncataloged books remaining after Monday.

75 books

Subtract the 45 cataloged books from the initial 120 books.

3

Calculate the number of books cataloged on Tuesday.

25 books

Multiply the remaining 75 books by the fraction cataloged on Tuesday, which is 1/3.

4

Calculate the final number of uncataloged books.

50 books

Subtract the 25 books cataloged on Tuesday from the 75 uncataloged books remaining after Monday.

Key Concept

Fraction of a Remaining Quantity

Estimated Time:50s

Question 64Question →

What is the largest two-digit positive integer $n$ such that when $n$ is divided by $4$ , the remainder is $3$ , and when $n$ is divided by $5$ , the remainder is $2$ ?

Show answer & explanation

Answer: 87

Answer

The largest two-digit positive integer satisfying both remainder conditions is 87.

Any integer satisfying both remainder requirements must be of the form

n = 20m + 7

for an integer

m

. Testing values for

m

shows that

m = 4

yields

n = 87

, which is the largest two-digit integer fitting this rule.

Step-by-Step Solution

1

Set up modular arithmetic equations for the given remainder conditions.

n \equiv 3 \pmod 4

and

n \equiv 2 \pmod 5

Translate the verbal description of remainders into mathematical congruence relations.

2

Combine the congruence relations to find the general form of

n

.

n = 20m + 7

for non-negative integers

m

The least common multiple of

4

and

5

is

20

, meaning solutions repeat every

20

units.

3

Find the maximum integer

m

that produces a two-digit integer.

For

m = 4

,

n = 87

. For

m = 5

,

n = 107

.

Two-digit integers are strictly less than

100

.

Key Concept

Simultaneous Remainders and Divisibility Cycles

Estimated Time:1m 30s

Question 65Question →

In a manufacturing facility, three automated machines—Machine A, Machine B, and Machine C—produce components at constant individual rates. Machine A working alone completes $\frac{3}{10}$ of a standard daily order in $3$ hours. Machine B working alone completes $\frac{2}{5}$ of the order in $6$ hours. Machine C working alone completes $\frac{1}{4}$ of the order in $5$ hours.

Initially, all three machines work together for $3$ hours. At the end of $3$ hours, Machine A malfunctions and stops, while Machines B and C continue working together without interruption until the daily order is finished. How many total hours does it take from the start to complete the entire daily order?

Show answer & explanation

Answer: 6

Answer

The total time required from the start to complete the entire daily order is 6 hours.

Each machine's rate per hour is found by dividing the given fraction by hours: Machine A is

\frac{1}{10}

, B is

\frac{1}{15}

, and C is

\frac{1}{20}

. Summing these gives an initial combined rate of

\frac{13}{60}

per hour. In the first 3 hours, the three machines complete

3 \times \frac{13}{60} = \frac{13}{20}

of the order, leaving

1 - \frac{13}{20} = \frac{7}{20}

of the order remaining. When Machine A stops, the remaining combined rate of B and C is

\frac{1}{15} + \frac{1}{20} = \frac{7}{60}

per hour. Dividing the remaining

\frac{7}{20}

by

\frac{7}{60}

yields 3 additional hours. Adding the initial 3 hours gives a total duration of 6 hours.

Step-by-Step Solution

1

Determine individual hourly work rates for each machine.

Machine A rate =

\frac{1}{10}

order/hr, Machine B rate =

\frac{1}{15}

order/hr, Machine C rate =

\frac{1}{20}

order/hr.

Divide the fraction of the job completed by the duration in hours to find the unit rate for each machine.

2

Calculate the combined rate of all three machines and the fraction completed in the first 3 hours.

Combined rate =

\frac{13}{60}

order/hr; Work completed in 3 hours =

\frac{13}{20}

, leaving

\frac{7}{20}

of the order unfinished.

Sum the three rates using a common denominator of 60, then multiply by 3 hours to find the total work done during the first stage.

3

Calculate the combined rate of Machines B and C, and find the additional time required to complete the remaining fraction.

Combined rate of B and C =

\frac{7}{60}

order/hr; Additional time needed = 3 hours.

Divide the remaining fraction

\frac{7}{20}

by the combined rate

\frac{7}{60}

to determine the remaining hours needed.

4

Add the initial duration and additional duration to obtain the total time.

Total time =

3 + 3 = 6

hours.

The question asks for the total elapsed time from the start of the job.

Key Concept

Adding rational fractions with different denominators to solve multi-stage work and rate problems.

Question 66Question →

An employee's hourly wage increased from $\$20.00$ to $\$25.00$ . What was the percent increase in the employee's hourly wage?

Show answer & explanation

Answer: 25

Answer

The percent increase in the employee's hourly wage is

25\%

.

The percent increase is calculated using the formula

\text{Percent Increase} = \frac{\text{New Value} - \text{Original Value}}{\text{Original Value}} \times 100\%

. Substituting the given values gives

\frac{25 - 20}{20} \times 100\% = \frac{5}{20} \times 100\% = 25\%

.

Step-by-Step Solution

1

Calculate the absolute increase in hourly wage

Increase =

\$25.00 - \$20.00 = \$5.00

Percent change is calculated relative to the absolute change from the initial value.

2

Calculate the percent change using the original wage as the base

\text{Percent Increase} = \frac{\$5.00}{\$20.00} \times 100\% = 25\%

The formula for percent change is

\frac{\text{Amount of Change}}{\text{Original Amount}} \times 100\%

.

Key Concept

Percent Change Formula

Question 67Question →

If $m$ is an even integer and $n$ is an odd integer, what is the value of $(-1)^m + (-1)^n$ ?

Show answer & explanation

Answer: 0

Answer

0

Since

m

is an even integer,

(-1)^m = 1

. Since

n

is an odd integer,

(-1)^n = -1

. Adding

1

and

-1

yields

0

.

Step-by-Step Solution

1

Evaluate

(-1)^m

for an even integer

m

(-1)^m = 1

Raising

-1

to an even integer power always results in

1

.

2

Evaluate

(-1)^n

for an odd integer

n

(-1)^n = -1

Raising

-1

to an odd integer power always results in

-1

.

3

Sum the two evaluated expressions

1 + (-1) = 0

Combining

1

and

-1

equals

0

.

Key Concept

Exponent sign rules for even and odd powers

Question 68Question →

In a community library, $\frac{4}{15}$ of the total book collection consists of fiction books, $\frac{2}{5}$ consists of non-fiction books, and the remaining 240 books are children's books. If $\frac{3}{8}$ of the fiction books are hardcovers and the rest are softcovers, how many softcover fiction books are in the library?

Show answer & explanation

Answer: 120

Answer

The total number of softcover fiction books in the library is 120.

To find the number of softcover fiction books, first calculate the total number of books in the library. The combined fraction of fiction and non-fiction books is

\frac{4}{15} + \frac{2}{5} = \frac{4}{15} + \frac{6}{15} = \frac{10}{15} = \frac{2}{3}

. Therefore, children's books represent the remaining

1 - \frac{2}{3} = \frac{1}{3}

of the collection. Since

\frac{1}{3}

of the total equals 240 books, the total collection is

240 \times 3 = 720

books. Next, compute the total number of fiction books:

\frac{4}{15} \times 720 = 192

. Finally, since

\frac{3}{8}

of fiction books are hardcovers, the remaining fraction of softcovers is

1 - \frac{3}{8} = \frac{5}{8}

. Multiplying

\frac{5}{8} \times 192

gives 120 softcover fiction books.

Step-by-Step Solution

1

Find the combined fraction of fiction and non-fiction books, then determine the fraction of children's books.

Combined fraction =

\frac{4}{15} + \frac{6}{15} = \frac{10}{15} = \frac{2}{3}

. Children's books fraction =

1 - \frac{2}{3} = \frac{1}{3}

.

The sum of all non-overlapping fractions comprising the entire collection must equal 1.

2

Calculate the total number of books in the library.

Total collection =

240 \div \frac{1}{3} = 720

books.

The 240 children's books represent exactly

\frac{1}{3}

of the whole collection.

3

Determine the total count of fiction books.

Fiction books =

\frac{4}{15} \times 720 = 192

books.

Fiction books make up

\frac{4}{15}

of the total 720 books.

4

Calculate the number of softcover fiction books.

Softcover fiction books =

\left(1 - \frac{3}{8}\right) \times 192 = \frac{5}{8} \times 192 = 120

books.

If

\frac{3}{8}

of the fiction subset are hardcovers, the remaining

\frac{5}{8}

of that subset are softcovers.

Key Concept

Solving multi-step rational number word problems by combining fractions and determining fractional parts of a subset.

Estimated Time:1m 30s

Question 69Question →

Two positive integers $a$ and $b$ have prime factorizations of the form $a = 2^x \times 3^y \times 5^z$ and $b = 2^u \times 3^v \times 5^w$ , where $x, y, z, u, v, w$ are non-negative integers. The greatest common divisor (GCD) of $a$ and $b$ is $180$ , and the least common multiple (LCM) of $a$ and $b$ is $54,000$ . If $a$ has exactly $24$ positive divisors, what is the value of $b$ ?

Show answer & explanation

Answer: 18000

Answer

18000

Prime factorizing

\text{GCD}(a,b) = 180 = 2^2 \times 3^2 \times 5^1

and

\text{LCM}(a,b) = 54,000 = 2^4 \times 3^3 \times 5^3

reveals that the pair of exponents for bases

2, 3, 5

are

\{2, 4\}

,

\{2, 3\}

, and

\{1, 3\}

respectively. Applying the total divisor formula

(x+1)(y+1)(z+1) = 24

for

a

,

x+1

must divide

24

. Since

x \in \{2, 4\}

,

x

must be

2

(since

5

does not divide

24

). This leaves

(y+1)(z+1) = 8

. With

y \in \{2, 3\}

,

y+1

must be

4

(since

3

does not divide

8

), giving

y = 3

and

z = 1

. Thus,

a = 2^2 \times 3^3 \times 5^1

. The remaining exponent choices for

b

are

u = 4

,

v = 2

, and

w = 3

, giving

b = 2^4 \times 3^2 \times 5^3 = 18,000

.

Step-by-Step Solution

1

Factorize GCD and LCM into prime bases.

GCD = 2^2 * 3^2 * 5^1; LCM = 2^4 * 3^3 * 5^3.

GCD represents the minimum prime exponents and LCM represents the maximum prime exponents of a and b.

2

Set up candidate exponent pairs for primes 2, 3, and 5.

{x, u} = {2, 4}, {y, v} = {2, 3}, and {z, w} = {1, 3}.

For any prime p, min(exp_p(a), exp_p(b)) = exp_p(GCD) and max(exp_p(a), exp_p(b)) = exp_p(LCM).

3

Solve for x, y, and z using the divisor count formula d(a) = (x+1)(y+1)(z+1) = 24.

x = 2, y = 3, z = 1.

If x = 4, then x+1 = 5, which is not a factor of 24. Thus x = 2. Then (y+1)(z+1) = 8. Since y in {2, 3}, y+1 = 4 implies y = 3 and z = 1.

4

Determine the exponents of b and calculate b.

u = 4, v = 2, w = 3, so b = 2^4 * 3^2 * 5^3 = 18,000.

Complementary values from the sets {x,u}, {y,v}, and {z,w} give the exponents for b.

Key Concept

Relationship between prime factor exponents in GCD, LCM, and divisor counting functions.

Question 70Question →

Let $N$ be a positive integer with exactly $12$ positive divisors. If the sum of the distinct prime factors of $N$ is $12$ and $N$ is not divisible by $4$ , what is the least possible value of $N$ ?

Show answer & explanation

Answer: 126

Answer

126

The least possible value of

N

is 126. The only sets of distinct prime factors summing to 12 are

\{5, 7\}

and

\{2, 3, 7\}

. Using

\{2, 3, 7\}

produces smaller candidates. Because

N

is not divisible by 4, the exponent of 2 must be 1. The total divisor condition

(1+1)(b+1)(c+1) = 12

requires

(b+1)(c+1) = 6

, giving exponent pairs

(1, 2)

or

(2, 1)

for bases 3 and 7. Assigning the exponent 2 to 3 and 1 to 7 minimizes

N

, giving

2^1 \cdot 3^2 \cdot 7^1 = 126

.

Step-by-Step Solution

1

Find all sets of distinct prime factors that sum to 12.

The possible sets of distinct prime factors are

\{5, 7\}

and

\{2, 3, 7\}

.

Testing combinations of prime numbers (

2, 3, 5, 7, 11, \dots

):

5 + 7 = 12

and

2 + 3 + 7 = 12

are the only valid sets of distinct primes summing to 12.

2

Analyze the set of prime factors

\{2, 3, 7\}

under the condition that

N

is not divisible by 4.

The prime factorization is

N = 2^a \cdot 3^b \cdot 7^c

, where

a = 1

.

Since 2 is a prime factor of

N

,

a \ge 1

. Because

N

is not divisible by 4 (

2^2

),

a

must be strictly less than 2. Thus,

a = 1

.

3

Determine the exponents

b

and

c

using the total number of positive divisors.

The number of divisors is

(1+1)(b+1)(c+1) = 12

, which simplifies to

(b+1)(c+1) = 6

. The possible integer pairs

(b, c)

for

b, c \ge 1

are

(1, 2)

and

(2, 1)

.

The divisor count formula for

N = p_1^{e_1} p_2^{e_2} \dots p_k^{e_k}

is

(e_1+1)(e_2+1)\dots(e_k+1) = 12

.

4

Calculate values of

N

for these exponent pairs.

For

(b, c) = (1, 2)

,

N = 2^1 \cdot 3^1 \cdot 7^2 = 294

. For

(b, c) = (2, 1)

,

N = 2^1 \cdot 3^2 \cdot 7^1 = 126

.

To make

N

as small as possible, assign the larger exponent to the smaller prime base (

3^2 \cdot 7^1 < 3^1 \cdot 7^2

).

5

Compare with candidate values from the alternative set of prime factors

\{5, 7\}

.

For

\{5, 7\}

,

N = 5^3 \cdot 7^2 = 6125

or

5^5 \cdot 7^1 = 21875

, both of which are much larger than 126.

The combination

(a+1)(b+1) = 12

yields exponents of 3 and 2 (or 5 and 1), resulting in much higher prime powers.

Key Concept

Divisor count formula combined with prime factorization constraints

Question 71Question →

If $|x - 4| = 9$ and $x < 0$ , what is the value of $x$ ?

Show answer & explanation

Answer: -5

Answer

The value of

x

is

-5

.

The absolute value equation

|x - 4| = 9

specifies that the distance between

x

and

4

on the real number line is equal to

9

. Moving

9

units to the left of

4

gives

4 - 9 = -5

, and moving

9

units to the right gives

4 + 9 = 13

. Because

x

is specified to be negative (

x < 0

), the correct value of

x

is

-5

.

Step-by-Step Solution

1

Set up the two linear equations representing the absolute value equation

|x - 4| = 9

.

x - 4 = 9

or

x - 4 = -9

By definition,

|A| = B

(where

B \ge 0

) implies

A = B

or

A = -B

.

2

Solve for

x

in both equations.

x = 13

or

x = -5

Adding

4

to both sides of

x - 4 = 9

gives

x = 13

, and adding

4

to both sides of

x - 4 = -9

gives

x = -5

.

3

Select the value of

x

that satisfies the given condition

x < 0

.

x = -5

The value

13

is positive, while

-5

is negative and fulfills

x < 0

.

Key Concept

Absolute Value as Distance and Solving Absolute Value Equations

Estimated Time:45s

Question 72Question →

At the beginning of the year, a cloud storage service had a certain number of active paid subscriptions. During the first six months of the year, the number of active subscriptions increased by $20\%$ . During the second six months of the year, the number of active subscriptions decreased by $15\%$ relative to the total at the middle of the year. If the service had $12,240$ active subscriptions at the end of the year, what was the number of active subscriptions at the beginning of the year?

Show answer & explanation

Answer: 12000

Answer

12000

To find the initial subscription count, model the successive percentage changes sequentially. Increasing the initial amount

N

by

20\%

yields

1.20N

. Decreasing that intermediate total by

15\%

multiplies

1.20N

by

0.85

, giving

1.02N

. Setting

1.02N = 12,240

gives

N = 12,000

.

Step-by-Step Solution

1

Express the mid-year subscription total in terms of the initial subscription count

N

.

Mid-year subscriptions

= N \times (1 + 0.20) = 1.20N

A

20\%

increase scales the original quantity by a factor of

1.20

.

2

Express the end-of-year subscription total in terms of

N

after a

15\%

decrease from the mid-year total.

End-of-year subscriptions

= 1.20N \times (1 - 0.15) = 1.20N \times 0.85 = 1.02N

A

15\%

decrease reduces the new base quantity to

85\%

of its mid-year value.

3

Set the algebraic expression for the final total equal to the given numerical total and solve for

N

.

1.02N = 12,240 \implies N = 12,000

Dividing the final amount by the net multiplier

1.02

yields the original starting value.

Key Concept

Successive Percentage Changes and Base Shift

Estimated Time:1m 30s

Question 73Question →

If $2^x \times 4^3 = 2^9$ , what is the value of $x$ ?

Show answer & explanation

Answer: 3

Answer

The value of

x

is 3.

By rewriting

4^3

as

(2^2)^3 = 2^6

, the equation becomes

2^x \cdot 2^6 = 2^9

. Applying the product rule gives

2^{x+6} = 2^9

, which simplifies to

x + 6 = 9

, yielding

x = 3

.

Step-by-Step Solution

1

Convert all terms to base 2

4^3 = (2^2)^3 = 2^{2 \times 3} = 2^6

To apply exponent rules, expressions should share a common base.

2

Apply product rule of exponents to the left side of the equation

2^x \times 2^6 = 2^{x+6}

When multiplying exponential terms with the same base, add their exponents:

a^m \cdot a^n = a^{m+n}

.

3

Equate the exponents from both sides

x + 6 = 9

Since the bases are identical and non-zero, the exponents must be equal.

4

Solve the linear equation for

x

x = 3

Subtract 6 from both sides of the equation.

Key Concept

Combining exponential terms with equal bases using power rules (

a^m \cdot a^n = a^{m+n}

and

(a^m)^n = a^{mn}

).

Question 74Question →

A high-precision instrument measures the mass of a single micro-particle $P$ as $4.8 \times 10^{-8}$ grams and a single micro-particle $Q$ as $8.0 \times 10^{-9}$ grams. A sample consists of a combination of $P$ and $Q$ particles in a ratio of $3$ particles of $P$ for every $5$ particles of $Q$ . If the total mass of the sample is $9.2 \times 10^{-5}$ grams, what is the total number of particles in the sample?

Show answer & explanation

Answer: 4000

Answer

The total number of particles in the sample is 4,000.

Aligning powers of 10 shows that one P particle weighs

4.8 \times 10^{-8}

g and one Q particle weighs

0.8 \times 10^{-8}

g. A combined unit of 3 P particles and 5 Q particles has a mass of

3(4.8 \times 10^{-8}) + 5(0.8 \times 10^{-8}) = 18.4 \times 10^{-8} = 1.84 \times 10^{-7}

g. Dividing the total sample mass of

9.2 \times 10^{-5}

g by

1.84 \times 10^{-7}

g yields 500 units. Since each unit contains 8 particles (3 + 5), the total number of particles is

500 \times 8 = 4,000

.

Step-by-Step Solution

1

Convert the mass of particle Q so that it shares the same exponent (

10^{-8}

) as particle P.

Mass of single particle Q =

0.8 \times 10^{-8}

grams.

Aligning powers of 10 is necessary before performing addition of masses.

2

Find the combined mass of a fundamental ratio group consisting of 3 particles of P and 5 particles of Q.

Mass of one ratio group =

3(4.8 \times 10^{-8}) + 5(0.8 \times 10^{-8}) = 14.4 \times 10^{-8} + 4.0 \times 10^{-8} = 18.4 \times 10^{-8} = 1.84 \times 10^{-7}

grams.

Determines the mass contributed by each set of 8 particles.

3

Divide the total mass of the sample by the mass of a single ratio group.

Number of groups =

\frac{9.2 \times 10^{-5}}{1.84 \times 10^{-7}} = \frac{9.2}{1.84} \times 10^2 = 5 \times 10^2 = 500

groups.

Determines how many full ratio sets of particles make up the sample.

4

Multiply the number of groups by the total number of particles contained in each group (

3 + 5 = 8

).

Total particles =

500 \times 8 = 4000

.

Yields the total count of individual particles in the sample.

Key Concept

Operations with scientific notation, decimal place value alignment, and weighted proportional sums.

Question 75Question →

If $x$ is a real number that satisfies the equation $\sqrt{x + 6\sqrt{x - 9}} + \sqrt{x - 6\sqrt{x - 9}} = 10$ , what is the value of $x$ ?

Show answer & explanation

Answer: 34

Answer

34

Using the substitution

u = \sqrt{x - 9} \ge 0

, we have

x = u^2 + 9

. The expressions under the square roots become

x + 6\sqrt{x - 9} = u^2 + 6u + 9 = (u + 3)^2

and

x - 6\sqrt{x - 9} = u^2 - 6u + 9 = (u - 3)^2

. Taking square roots gives

\sqrt{(u + 3)^2} + \sqrt{(u - 3)^2} = (u + 3) + |u - 3| = 10

. For

u \ge 3

, this simplifies to

(u + 3) + (u - 3) = 10 \implies 2u = 10 \implies u = 5

. Finally, substituting

u = 5

back yields

x = 5^2 + 9 = 34

.

Step-by-Step Solution

1

Define a variable substitution to simplify the nested radical structure.

Let

u = \sqrt{x - 9}

where

u \ge 0

. Squaring both sides gives

u^2 = x - 9

, so

x = u^2 + 9

.

This substitution allows the expressions inside the outer square roots to be rewritten as polynomials in terms of

u

.

2

Rewrite the expressions under each square root as perfect square trinomials.

x + 6\sqrt{x - 9} = (u^2 + 9) + 6u = (u + 3)^2

and

x - 6\sqrt{x - 9} = (u^2 + 9) - 6u = (u - 3)^2

.

Expressing terms as perfect squares allows the outer radicals to be simplified.

3

Simplify the square root expressions using absolute values.

\sqrt{(u + 3)^2} + \sqrt{(u - 3)^2} = (u + 3) + |u - 3| = 10

.

For any real number

a

,

\sqrt{a^2} = |a|

. Since

u \ge 0

,

u + 3 > 0

, so

|u + 3| = u + 3

.

4

Solve the absolute value equation across valid domain intervals.

If

u \ge 3

,

|u - 3| = u - 3

, giving

(u + 3) + (u - 3) = 10 \implies 2u = 10 \implies u = 5

. If

0 \le u < 3

,

|u - 3| = 3 - u

, giving

(u + 3) + (3 - u) = 6 \neq 10

(no solution). Thus,

u = 5

.

Splitting into cases based on the definition of absolute value isolates the valid root.

5

Substitute

u = 5

back into the expression for

x

.

x = 5^2 + 9 = 25 + 9 = 34

.

Converting from

u

back to

x

provides the solution to the original equation.

Key Concept

Simplifying nested radicals by completing the square under the radical sign and applying the identity

\sqrt{a^2} = |a|

.

Question 76Question →

What is the least integer value of $x$ that satisfies the inequality $|4x - 5| \le 11$ ?

Show answer & explanation

Answer: -1

Answer

The least integer value of

x

that satisfies the inequality is

-1

.

Expanding the absolute value inequality

|4x - 5| \le 11

produces

-11 \le 4x - 5 \le 11

. Adding 5 across all sections yields

-6 \le 4x \le 16

, and dividing by 4 results in

-1.5 \le x \le 4

. The set of integers within this range is

\{-1, 0, 1, 2, 3, 4\}

, making

-1

the least integer value.

Step-by-Step Solution

1

Rewrite the absolute value inequality as a double inequality.

-11 \le 4x - 5 \le 11

An absolute value inequality of the form

|u| \le a

for

a \ge 0

is equivalent to

-a \le u \le a

.

2

Add 5 to all parts of the compound inequality.

-6 \le 4x \le 16

To isolate

x

, first add 5 to eliminate the constant term

-5

.

3

Divide all parts by 4.

-1.5 \le x \le 4

Dividing by a positive constant preserves the direction of the inequality signs.

4

Select the minimum integer contained within the solution interval

[-1.5, 4]

.

-1

The integer values satisfying

-1.5 \le x \le 4

are

-1, 0, 1, 2, 3, 4

. The smallest among these is

-1

.

Key Concept

Linear inequalities involving absolute value

Question 77Question →

A coffee shop owner creates a $30$ -pound blend of coffee by mixing Bean X, which costs $\$8$ per pound, with Bean Y, which costs $\$14$ per pound. If the final mixture costs $\$10$ per pound, how many pounds of Bean X are in the mixture?

Show answer & explanation

Answer: 20

Answer

The mixture contains 20 pounds of Bean X.

The quantity of Bean X required is 20 pounds. Setting up the cost equation

8x + 14(30 - x) = 300

simplifies to

-6x = -120

, giving

x = 20

.

Step-by-Step Solution

1

Define the unknown variables representing the quantities of each component.

Let

x

be the weight of Bean X in pounds. The weight of Bean Y is

(30 - x)

pounds.

Since the total weight of the mixture is 30 pounds, expressing Bean Y in terms of

x

creates a single-variable system.

2

Set up a linear equation based on total financial value.

8x + 14(30 - x) = 10 \times 30

The total cost of Bean X plus the total cost of Bean Y equals the total cost of the combined 30-pound mixture.

3

Solve the algebraic equation for

x

.

8x + 420 - 14x = 300 \implies -6x = -120 \implies x = 20

Combining like terms isolates the variable

x

to yield the required quantity of Bean X.

Key Concept

Linear mixture modeling and single-variable algebraic modeling

Question 78Question →

Circle $C_1$ has radius $r_1$ and Circle $C_2$ has radius $r_2$ . An arc on Circle $C_1$ subtended by a central angle of $\theta_1^\circ$ has the exact same length as an arc on Circle $C_2$ subtended by a central angle of $\theta_2^\circ$ . The sector formed by this arc in Circle $C_1$ has an area of $54\pi$ , and the sector formed by this arc in Circle $C_2$ has an area of $36\pi$ . If $\theta_1 + \theta_2 = 150$ , what is the value of $r_1$ ?

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Answer: 18

Answer

The radius

r_1

of Circle

C_1

is 18.

Using the relationship

A = \frac{1}{2} L r

, the ratio of the two sector areas gives

\frac{r_1}{r_2} = \frac{54\pi}{36\pi} = 1.5

, so

r_2 = \frac{2}{3} r_1

. Expressing arc length as

L = \frac{108\pi}{r_1}

, the central angles are

\theta_1 = \frac{19440}{r_1^2}

and

\theta_2 = \frac{29160}{r_1^2}

. Setting their sum equal to

150

leads to

\frac{48600}{r_1^2} = 150

, giving

r_1^2 = 324

and

r_1 = 18

.

Step-by-Step Solution

1

Relate sector area to arc length and radius

A_1 = \frac{1}{2} L r_1 = 54\pi

and

A_2 = \frac{1}{2} L r_2 = 36\pi

The area of a sector with arc length

L

and radius

r

is given by

A = \frac{\theta}{360}\pi r^2 = \frac{1}{2} L r

.

2

Find the ratio of

r_1

to

r_2

\frac{r_1}{r_2} = \frac{54\pi}{36\pi} = \frac{3}{2} \implies r_2 = \frac{2}{3}r_1

Dividing the first area equation by the second cancels out

\frac{1}{2}L

.

3

Express central angles in terms of

r_1

\theta_1 = \frac{19440}{r_1^2}

and

\theta_2 = \frac{29160}{r_1^2}

Since

L = \frac{108\pi}{r_1}

, substituting into

\theta_1 = \frac{L}{2\pi r_1} \times 360

yields

\theta_1 = \frac{19440}{r_1^2}

, and substituting into

\theta_2 = \frac{L}{2\pi (\frac{2}{3}r_1)} \times 360

yields

\theta_2 = \frac{29160}{r_1^2}

.

4

Solve for

r_1

using the angle sum equation

r_1 = 18

Summing the angles gives

\frac{19440 + 29160}{r_1^2} = \frac{48600}{r_1^2} = 150 \implies r_1^2 = 324 \implies r_1 = 18

.

Key Concept

Relationship between arc length, radius, central angle, and sector area

Estimated Time:2m 30s

Question 79Question →

If $x$ and $y$ are real numbers that satisfy the inequalities $|3 - 2x| \le 9$ and $|4y + 3| \le 13$ , what is the maximum possible value of the expression $|3x - 4y|$ ?

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Answer: 34

Answer

The maximum possible value of

|3x - 4y|

is

34

.

To find the maximum possible value of

|3x - 4y|

, we first solve for the ranges of

x

and

y

from their respective absolute value inequalities. From

|3 - 2x| \le 9

, we get

-9 \le 3 - 2x \le 9 \implies -3 \le x \le 6

. From

|4y + 3| \le 13

, we get

-13 \le 4y + 3 \le 13 \implies -4 \le y \le 2.5

. To maximize

3x - 4y

, we take the maximum value of

3x

(

3 \times 6 = 18

) and the maximum value of

-4y

(

-4 \times -4 = 16

), giving

18 + 16 = 34

. To minimize

3x - 4y

, we take the minimum value of

3x

(

3 \times -3 = -9

) and the minimum value of

-4y

(

-4 \times 2.5 = -10

), giving

-9 - 10 = -19

. The expression

3x - 4y

ranges from

-19

to

34

, so the maximum possible magnitude

|3x - 4y|

is

\max(|-19|, |34|) = 34

.

Step-by-Step Solution

1

Solve the inequality

|3 - 2x| \le 9

for

x

.

-3 \le x \le 6

Unpacking the absolute value yields

-9 \le 3 - 2x \le 9

. Subtracting

3

gives

-12 \le -2x \le 6

. Dividing by

-2

and reversing the inequality signs produces

-3 \le x \le 6

.

2

Solve the inequality

|4y + 3| \le 13

for

y

.

-4 \le y \le 2.5

Unpacking the absolute value yields

-13 \le 4y + 3 \le 13

. Subtracting

3

gives

-16 \le 4y \le 10

. Dividing by

4

gives

-4 \le y \le 2.5

.

3

Find the range of possible values for

3x

and

-4y

.

-9 \le 3x \le 18

and

-10 \le -4y \le 16

Multiplying

-3 \le x \le 6

by

3

gives

-9 \le 3x \le 18

. Multiplying

-4 \le y \le 2.5

by

-4

and flipping the signs gives

-10 \le -4y \le 16

.

4

Combine the bounds for

3x

and

-4y

to find the range for

3x - 4y

.

-19 \le 3x - 4y \le 34

The minimum value of

3x - 4y

is

(-9) + (-10) = -19

. The maximum value of

3x - 4y

is

18 + 16 = 34

.

5

Determine the maximum absolute value

|3x - 4y|

over the interval

[-19, 34]

.

34

The absolute value of any number in the interval

[-19, 34]

ranges from

0

to

\max(|-19|, |34|) = 34

.

Key Concept

Absolute Value Inequalities and Expression Bounding

Question 80Question →

Lines $L_1$ and $L_2$ intersect at point $P$ to form an acute angle of $54^\circ$ . Line $M$ passes through point $P$ and is perpendicular to line $L_1$ . Ray $R$ originates at point $P$ and lies in the interior of one of the obtuse angles formed by $L_1$ and $L_2$ . If Ray $R$ bisects the angle formed between line $M$ and line $L_2$ , what is the measure, in degrees, of the acute angle formed by Ray $R$ and line $L_1$ ?

Show answer & explanation

Answer: 72

Answer

The measure of the acute angle formed by Ray

R

and line

L_1

is

72^\circ

.

Lines

L_1

and

L_2

form an obtuse angle of

180^\circ - 54^\circ = 126^\circ

. Perpendicular line

M

consumes

90^\circ

of this angle relative to

L_1

, leaving

36^\circ

between line

M

and line

L_2

. Bisecting this remaining angle gives

18^\circ

. Ray

R

therefore lies

18^\circ

away from perpendicular line

M

, making an acute angle of

90^\circ - 18^\circ = 72^\circ

with line

L_1

.

Step-by-Step Solution

1

Determine the measure of the obtuse angle between lines

L_1

and

L_2

.

The obtuse angle measures

180^\circ - 54^\circ = 126^\circ

.

Adjacent angles along intersecting straight lines are supplementary and sum to

180^\circ

.

2

Find the angle between perpendicular line

M

and line

L_2

.

The angle between line

M

and line

L_2

within the obtuse region is

126^\circ - 90^\circ = 36^\circ

.

Line

M

is perpendicular to line

L_1

, taking up

90^\circ

of the

126^\circ

obtuse angle.

3

Determine the angle formed by Ray

R

after bisecting the

36^\circ

angle.

The angle between Ray

R

and line

M

is

36^\circ / 2 = 18^\circ

.

An angle bisector divides an angle into two equal parts.

4

Calculate the acute angle between Ray

R

and line

L_1

.

The acute angle formed between Ray

R

and line

L_1

is

90^\circ - 18^\circ = 72^\circ

.

Line

M

forms a

90^\circ

angle with line

L_1

. Subtracting the

18^\circ

offset created by Ray

R

yields the acute angle of

72^\circ

.

Key Concept

Supplementary angles, perpendicular lines, and angle bisectors

Estimated Time:2m 0s

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